Prove that a cyclic group with only one generator can have at most 2 elements
$begingroup$
Prove that a cyclic group that has only one generator has at most $2$ elements.
I want to know if my proof would be valid:
Suppose $G$ is a cyclic group and $g$ is its only generator. Let $|G|=n$ where $n>2$, then we know that $gcd(n,n-1)=1$. This implies that $g^{n-1}$ is a generator of $G$. We have a contradiction, since $g$ is the only generator of $G$ (and $n > 2$ leads to $n-1 neq 1$). Thus $|G|leq 2$.
I tried to use the fact that generating elements of a group are coprime to the order of the group, thanks.
abstract-algebra greatest-common-divisor cyclic-groups
New contributor
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add a comment |
$begingroup$
Prove that a cyclic group that has only one generator has at most $2$ elements.
I want to know if my proof would be valid:
Suppose $G$ is a cyclic group and $g$ is its only generator. Let $|G|=n$ where $n>2$, then we know that $gcd(n,n-1)=1$. This implies that $g^{n-1}$ is a generator of $G$. We have a contradiction, since $g$ is the only generator of $G$ (and $n > 2$ leads to $n-1 neq 1$). Thus $|G|leq 2$.
I tried to use the fact that generating elements of a group are coprime to the order of the group, thanks.
abstract-algebra greatest-common-divisor cyclic-groups
New contributor
$endgroup$
add a comment |
$begingroup$
Prove that a cyclic group that has only one generator has at most $2$ elements.
I want to know if my proof would be valid:
Suppose $G$ is a cyclic group and $g$ is its only generator. Let $|G|=n$ where $n>2$, then we know that $gcd(n,n-1)=1$. This implies that $g^{n-1}$ is a generator of $G$. We have a contradiction, since $g$ is the only generator of $G$ (and $n > 2$ leads to $n-1 neq 1$). Thus $|G|leq 2$.
I tried to use the fact that generating elements of a group are coprime to the order of the group, thanks.
abstract-algebra greatest-common-divisor cyclic-groups
New contributor
$endgroup$
Prove that a cyclic group that has only one generator has at most $2$ elements.
I want to know if my proof would be valid:
Suppose $G$ is a cyclic group and $g$ is its only generator. Let $|G|=n$ where $n>2$, then we know that $gcd(n,n-1)=1$. This implies that $g^{n-1}$ is a generator of $G$. We have a contradiction, since $g$ is the only generator of $G$ (and $n > 2$ leads to $n-1 neq 1$). Thus $|G|leq 2$.
I tried to use the fact that generating elements of a group are coprime to the order of the group, thanks.
abstract-algebra greatest-common-divisor cyclic-groups
abstract-algebra greatest-common-divisor cyclic-groups
New contributor
New contributor
edited 15 hours ago
darij grinberg
11.3k33164
11.3k33164
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asked 16 hours ago
PabloPablo
633
633
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3 Answers
3
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oldest
votes
$begingroup$
Perhaps easier: if $g$ generates $G$, then so does $g^{-1}$. The hypothesis then implies that $g=g^{-1}$, so $g^2=1$. Done (either $g=1$ or not, in which case respectively the order of $G$ is $1$ or $2$).
$endgroup$
$begingroup$
Beautiful! Thank you so much for this!
$endgroup$
– Pablo
8 hours ago
2
$begingroup$
When $g$ is a generator of a cyclic group of order $n$, then $g^{-1}$ is just another way to write $g^{n-1}$, so this is not really all that different from the proof proposed in the question. Though indeed by avoiding to mention the order of the cyclic group, this answer does handle the infinite cyclic group where the one in the question overlooks that possibiliy.
$endgroup$
– Marc van Leeuwen
3 hours ago
add a comment |
$begingroup$
Your proof is correct if $G$ is finite, i.e. $Gcongmathbb{Z}_m$ for some $mge 1$. Just notice that it may happen that $Gcongmathbb{Z}$; however, in this case $g$ and $g^{-1}$ are distinct generators and this concludes the proof.
$endgroup$
$begingroup$
That's a very good observation, thanks!
$endgroup$
– Pablo
8 hours ago
add a comment |
$begingroup$
Here is another take.
The number of generators is $phi(n)$, where $phi$ is Euler's function.
Now, either $n$ has a prime factor $pge 3$ or $n$ is a power of $2$.
In the first case, we have $phi(n) ge phi(p)=p-1ge2$.
In the second case, if $nge 3$, then $4$ divides $n$ and so $phi(n) ge phi(4)=2$.
Bottom line, $phi(n)=1$ implies $n$ is a power of $2$ less than $4$, that is, $n=1$ or $n=2$.
(The key fact is: if $d$ divides $n$, then $phi(n) ge phi(d)$.)
$endgroup$
1
$begingroup$
The argument in the very last sentence is wrong (though the claim is true).
$endgroup$
– darij grinberg
14 hours ago
$begingroup$
@darij, thanks.
$endgroup$
– lhf
5 hours ago
add a comment |
Your Answer
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Perhaps easier: if $g$ generates $G$, then so does $g^{-1}$. The hypothesis then implies that $g=g^{-1}$, so $g^2=1$. Done (either $g=1$ or not, in which case respectively the order of $G$ is $1$ or $2$).
$endgroup$
$begingroup$
Beautiful! Thank you so much for this!
$endgroup$
– Pablo
8 hours ago
2
$begingroup$
When $g$ is a generator of a cyclic group of order $n$, then $g^{-1}$ is just another way to write $g^{n-1}$, so this is not really all that different from the proof proposed in the question. Though indeed by avoiding to mention the order of the cyclic group, this answer does handle the infinite cyclic group where the one in the question overlooks that possibiliy.
$endgroup$
– Marc van Leeuwen
3 hours ago
add a comment |
$begingroup$
Perhaps easier: if $g$ generates $G$, then so does $g^{-1}$. The hypothesis then implies that $g=g^{-1}$, so $g^2=1$. Done (either $g=1$ or not, in which case respectively the order of $G$ is $1$ or $2$).
$endgroup$
$begingroup$
Beautiful! Thank you so much for this!
$endgroup$
– Pablo
8 hours ago
2
$begingroup$
When $g$ is a generator of a cyclic group of order $n$, then $g^{-1}$ is just another way to write $g^{n-1}$, so this is not really all that different from the proof proposed in the question. Though indeed by avoiding to mention the order of the cyclic group, this answer does handle the infinite cyclic group where the one in the question overlooks that possibiliy.
$endgroup$
– Marc van Leeuwen
3 hours ago
add a comment |
$begingroup$
Perhaps easier: if $g$ generates $G$, then so does $g^{-1}$. The hypothesis then implies that $g=g^{-1}$, so $g^2=1$. Done (either $g=1$ or not, in which case respectively the order of $G$ is $1$ or $2$).
$endgroup$
Perhaps easier: if $g$ generates $G$, then so does $g^{-1}$. The hypothesis then implies that $g=g^{-1}$, so $g^2=1$. Done (either $g=1$ or not, in which case respectively the order of $G$ is $1$ or $2$).
answered 16 hours ago
MPWMPW
30.2k12157
30.2k12157
$begingroup$
Beautiful! Thank you so much for this!
$endgroup$
– Pablo
8 hours ago
2
$begingroup$
When $g$ is a generator of a cyclic group of order $n$, then $g^{-1}$ is just another way to write $g^{n-1}$, so this is not really all that different from the proof proposed in the question. Though indeed by avoiding to mention the order of the cyclic group, this answer does handle the infinite cyclic group where the one in the question overlooks that possibiliy.
$endgroup$
– Marc van Leeuwen
3 hours ago
add a comment |
$begingroup$
Beautiful! Thank you so much for this!
$endgroup$
– Pablo
8 hours ago
2
$begingroup$
When $g$ is a generator of a cyclic group of order $n$, then $g^{-1}$ is just another way to write $g^{n-1}$, so this is not really all that different from the proof proposed in the question. Though indeed by avoiding to mention the order of the cyclic group, this answer does handle the infinite cyclic group where the one in the question overlooks that possibiliy.
$endgroup$
– Marc van Leeuwen
3 hours ago
$begingroup$
Beautiful! Thank you so much for this!
$endgroup$
– Pablo
8 hours ago
$begingroup$
Beautiful! Thank you so much for this!
$endgroup$
– Pablo
8 hours ago
2
2
$begingroup$
When $g$ is a generator of a cyclic group of order $n$, then $g^{-1}$ is just another way to write $g^{n-1}$, so this is not really all that different from the proof proposed in the question. Though indeed by avoiding to mention the order of the cyclic group, this answer does handle the infinite cyclic group where the one in the question overlooks that possibiliy.
$endgroup$
– Marc van Leeuwen
3 hours ago
$begingroup$
When $g$ is a generator of a cyclic group of order $n$, then $g^{-1}$ is just another way to write $g^{n-1}$, so this is not really all that different from the proof proposed in the question. Though indeed by avoiding to mention the order of the cyclic group, this answer does handle the infinite cyclic group where the one in the question overlooks that possibiliy.
$endgroup$
– Marc van Leeuwen
3 hours ago
add a comment |
$begingroup$
Your proof is correct if $G$ is finite, i.e. $Gcongmathbb{Z}_m$ for some $mge 1$. Just notice that it may happen that $Gcongmathbb{Z}$; however, in this case $g$ and $g^{-1}$ are distinct generators and this concludes the proof.
$endgroup$
$begingroup$
That's a very good observation, thanks!
$endgroup$
– Pablo
8 hours ago
add a comment |
$begingroup$
Your proof is correct if $G$ is finite, i.e. $Gcongmathbb{Z}_m$ for some $mge 1$. Just notice that it may happen that $Gcongmathbb{Z}$; however, in this case $g$ and $g^{-1}$ are distinct generators and this concludes the proof.
$endgroup$
$begingroup$
That's a very good observation, thanks!
$endgroup$
– Pablo
8 hours ago
add a comment |
$begingroup$
Your proof is correct if $G$ is finite, i.e. $Gcongmathbb{Z}_m$ for some $mge 1$. Just notice that it may happen that $Gcongmathbb{Z}$; however, in this case $g$ and $g^{-1}$ are distinct generators and this concludes the proof.
$endgroup$
Your proof is correct if $G$ is finite, i.e. $Gcongmathbb{Z}_m$ for some $mge 1$. Just notice that it may happen that $Gcongmathbb{Z}$; however, in this case $g$ and $g^{-1}$ are distinct generators and this concludes the proof.
answered 15 hours ago
LBJFSLBJFS
2307
2307
$begingroup$
That's a very good observation, thanks!
$endgroup$
– Pablo
8 hours ago
add a comment |
$begingroup$
That's a very good observation, thanks!
$endgroup$
– Pablo
8 hours ago
$begingroup$
That's a very good observation, thanks!
$endgroup$
– Pablo
8 hours ago
$begingroup$
That's a very good observation, thanks!
$endgroup$
– Pablo
8 hours ago
add a comment |
$begingroup$
Here is another take.
The number of generators is $phi(n)$, where $phi$ is Euler's function.
Now, either $n$ has a prime factor $pge 3$ or $n$ is a power of $2$.
In the first case, we have $phi(n) ge phi(p)=p-1ge2$.
In the second case, if $nge 3$, then $4$ divides $n$ and so $phi(n) ge phi(4)=2$.
Bottom line, $phi(n)=1$ implies $n$ is a power of $2$ less than $4$, that is, $n=1$ or $n=2$.
(The key fact is: if $d$ divides $n$, then $phi(n) ge phi(d)$.)
$endgroup$
1
$begingroup$
The argument in the very last sentence is wrong (though the claim is true).
$endgroup$
– darij grinberg
14 hours ago
$begingroup$
@darij, thanks.
$endgroup$
– lhf
5 hours ago
add a comment |
$begingroup$
Here is another take.
The number of generators is $phi(n)$, where $phi$ is Euler's function.
Now, either $n$ has a prime factor $pge 3$ or $n$ is a power of $2$.
In the first case, we have $phi(n) ge phi(p)=p-1ge2$.
In the second case, if $nge 3$, then $4$ divides $n$ and so $phi(n) ge phi(4)=2$.
Bottom line, $phi(n)=1$ implies $n$ is a power of $2$ less than $4$, that is, $n=1$ or $n=2$.
(The key fact is: if $d$ divides $n$, then $phi(n) ge phi(d)$.)
$endgroup$
1
$begingroup$
The argument in the very last sentence is wrong (though the claim is true).
$endgroup$
– darij grinberg
14 hours ago
$begingroup$
@darij, thanks.
$endgroup$
– lhf
5 hours ago
add a comment |
$begingroup$
Here is another take.
The number of generators is $phi(n)$, where $phi$ is Euler's function.
Now, either $n$ has a prime factor $pge 3$ or $n$ is a power of $2$.
In the first case, we have $phi(n) ge phi(p)=p-1ge2$.
In the second case, if $nge 3$, then $4$ divides $n$ and so $phi(n) ge phi(4)=2$.
Bottom line, $phi(n)=1$ implies $n$ is a power of $2$ less than $4$, that is, $n=1$ or $n=2$.
(The key fact is: if $d$ divides $n$, then $phi(n) ge phi(d)$.)
$endgroup$
Here is another take.
The number of generators is $phi(n)$, where $phi$ is Euler's function.
Now, either $n$ has a prime factor $pge 3$ or $n$ is a power of $2$.
In the first case, we have $phi(n) ge phi(p)=p-1ge2$.
In the second case, if $nge 3$, then $4$ divides $n$ and so $phi(n) ge phi(4)=2$.
Bottom line, $phi(n)=1$ implies $n$ is a power of $2$ less than $4$, that is, $n=1$ or $n=2$.
(The key fact is: if $d$ divides $n$, then $phi(n) ge phi(d)$.)
edited 5 hours ago
answered 15 hours ago
lhflhf
165k10171396
165k10171396
1
$begingroup$
The argument in the very last sentence is wrong (though the claim is true).
$endgroup$
– darij grinberg
14 hours ago
$begingroup$
@darij, thanks.
$endgroup$
– lhf
5 hours ago
add a comment |
1
$begingroup$
The argument in the very last sentence is wrong (though the claim is true).
$endgroup$
– darij grinberg
14 hours ago
$begingroup$
@darij, thanks.
$endgroup$
– lhf
5 hours ago
1
1
$begingroup$
The argument in the very last sentence is wrong (though the claim is true).
$endgroup$
– darij grinberg
14 hours ago
$begingroup$
The argument in the very last sentence is wrong (though the claim is true).
$endgroup$
– darij grinberg
14 hours ago
$begingroup$
@darij, thanks.
$endgroup$
– lhf
5 hours ago
$begingroup$
@darij, thanks.
$endgroup$
– lhf
5 hours ago
add a comment |
Pablo is a new contributor. Be nice, and check out our Code of Conduct.
Pablo is a new contributor. Be nice, and check out our Code of Conduct.
Pablo is a new contributor. Be nice, and check out our Code of Conduct.
Pablo is a new contributor. Be nice, and check out our Code of Conduct.
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